Hypothesis: The relationship between happiness and valence is moderated by music sophistication

Setup

To check this prediction we perform a moderation analysis. We keep the standardized version of the happiness scores because of the same reasons exposed in the simple linear regression.

Checking extra assumptions

Performing a moderation analysis implies entering the domains of the multiple linear regression (i.e. now there os more than one predictor). This means that the multicollinearity between predictors must be checked. To do so we look at the VIF values of the two predictors used, and there seems to be no multicollinearity at all.

               X   sophistication X:sophistication 
        1.004144         1.000172         1.004172 

Fitting the model

The model points at the interaction term being not significant and R2 = 0.1043. This points at our prediction not being met: the interaction is not significant and including this interaction term into the regression

Call:
lm(formula = Y ~ X + sophistication + X * sophistication)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.56948 -0.20703  0.01238  0.17519  0.53605 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       5.262e-01  6.956e-03  75.650   <2e-16 ***
X                 5.202e-02  4.413e-03  11.787   <2e-16 ***
sophistication    3.641e-05  3.944e-04   0.092    0.926    
X:sophistication -2.765e-04  2.407e-04  -1.149    0.251    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.2379 on 1166 degrees of freedom
Multiple R-squared:  0.1066,    Adjusted R-squared:  0.1043 
F-statistic: 46.36 on 3 and 1166 DF,  p-value: < 2.2e-16

Analysis and conclusion

Although the fit statistics already point at our prediction being not supported by the data, we can explore it visually. To do so we plot the valence distribution of songs at each level of happiness, for three different sophistication levels. If there was a moderation effect we would expect the direction of the relationship to change significantly in at least one of the three sophistication levels. Nevertheless, in all three plots, the relationship between valence and happiness seems identical.

Hypothesis: The variance in perceived happiness scores is lower among highly sophisticated participants

Sophistication spread

Levene’s test

To check this prediction we use the Levene’s test. In doing so we compare the happiness scores variance of the ‘low’,‘mid’ and ‘high’ sophistication group for each of the 15 songs. If the test shows a significant p-value, at least one of the groups has a different variance than the others. To see exactly what group has this different variance and visualize the data, we plot the happiness score distribution for each sophistication group using a boxplot. If our prediction was supported by the data, then the dark green box (i.e. variance of the highly sophisticated participants) would be systematically smaller than the lighter green boxes (i.e. variance of the ‘mid’ and ‘low’ participants). As a final note, Levene’s test’s p-values are corrected for multiplied testing using Bonferroni’s correction.

Analysis

As we can see from the corrected Levene’s tests p-value, none of the songs groups have different variances. From visual inspection we can also see that the dark green box does not tend to be smaller than the other two.

Considerations

It could be the case that our sample has a low sophistication in general. This would mean that our ‘high’ group still has a low sophistication in absolute terms, and thus we are not able to infer anything about highly sophisticated people and variance in happiness scores. To check this we plot the music sophistication distribution in our sample. We can see here nevertheless, that sophistication levels are sufficiently spread and we can consider our sample representative enough.